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it follows that for each point (xl9 . . . , xn) in E1 x    x En , we have

(             />'("(*!>    > *)) ^ * />i(*i) ' "AW-

Conversely, it is clear that if for each seminorm p' e F' there exist semi-
norms pjeTj and a real number c>0 such that ( is satisfied
throughout E1 x - x En9 then u is continuous at the point (0, . ..,0).
Now let (bl9\..9 bn) be an arbitrary point of Et x    x En , and write

u(xl9 ...,*)- w(^i>    A) =


where H runs through the 2" - 1 proper subsets of {1, 2, . . . , n}9 and for each
such H put yj = bj if j e H, and yj = Xj - bj if j $ H. Then it follows
immediately that u is continuous at the point (bl9 ..., bn), i.e., u is continuous

on E! x   * x En .

(12.14.12) On a vector space E, let F and F' be two families of seminorms.
Then the topology defined by F is finer than the topology defined by F' if and
only if, for each seminorm pr e F', there exists a finite family (Pth^n of semi-
norms belonging to F and a real number c > 0 such that p'(x) :g c  sup /?f(x)
/or all xeE.                                                                                  l

Apply (12.14.11) to the identity map u = 1E (12.2.1).

(1 2.1 4.1 3) Finally, we remark that the notions of series and convergent series
are defined in a topological vector space E just as in a normed space (5.2);
propositions (5.2.2), (5.2.3), (5.2.4) and the fact that the general term of a
convergent Series tends to 0 remain valid without change. If E is locally
convex and its topology is defined by a family (px) of seminorms, then for a
series with general term xn to converge in E it is necessary that, for each s > 0
and each scalar A, there should exist an integer /i0 such that

for all n ;> n0 and q > 0. If E is a Frechet space, this condition is also sufficient,
as follows from Cauchy's criterion (12.9).


1.   Let E be a vector space over R (resp. C) and let 58 be a set of subsets of E satisfying
conditions (VO and (Vn) of Section 12.3, Problem 3, together with the following:
(EVi)   Every set V e 95 is balanced and absorbing.   x En , and for each j let Xj > 0 be such